A GENERALIZATION OF THE CARTAN-KAHLER THEOREM.
WASHINGTON UNIV SEATTLE DEPT OF MATHEMATICS
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An existence theorem for hyperbolic systems of partial differential equations is used to obtain a generalization of the Cartan-Kahler theorem to non-analytic differential systems. In order to apply this theorem certain restrictions must be placed on the differential system considered. A Ck-differential system S in r independent variables satisfying these conditions is said to be Ckhyperbolic in the xr-direction. After making this definition the following theorem is proved. Let S be Ckhyperbolic in the xr-direction with k or 4r1 and genus g or r. Suppose I sub r-1 is an r-1-dimensional Ck-integral submanifold of S. Then in a neighborhood of each regular point p contained in I sub r-1 of S there exists an r-dimensional Ck-integral submanifold containing I sub r-1. In case r 2 the differential system need only be C1 for the theorem to hold. The concept of a moving frame is introduced and used to set up four applications of the above theorem to problems in surface theory. In the section on applications it is proved that locally a 2-dimensional Ck-Riemannian manifold has a Ck-isometric imbedding in 3-dimensional euclidean space for k or 1. It is shown that the theorem cannot be used to obtain a local Ck-conformal equivalence between two surfaces. Author